Efficient Circuits for Quantum Search over 2D Square Lattice Architecture

Quantum computing has increasingly drawn interest and investments from the academic, industrial, and governmental research communities worldwide. Among quantum algorithms, Quantum Search is important for its quadratic speedup over its classical-computing counterpart. A key ingredient in its implementation is the Multi-Control Toffoli (MCT) gate, which creates a Boolean product of control variables and XORs it into the target. On an idealized quantum computer, all-to-all connectivity would eliminate the need to use SWAP gates to communicate information. This is, however, not affordable in the current Noisy Intermediate-Scale Quantum (NISQ) computing era. In this work, we discuss how to efficiently implement MCT gates on 2D Square Lattices (2DSL), suitable for superconducting circuits, by taking advantage of relative-phase Toffoli gates and H-tree layouts to drastically reduce resulting circuits' depths and the amount of SWAPping required.